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\write 0 {5.  TRANSFORMATION BETWEEN THE CELESTIAL AND TERRESTRIAL
SYSTEMS \noexpand\dotfill\the\pageno\noexpand\break}
\hangindent=7.2pc
\hangafter=1
\noindent{\large CHAPTER 5} {\bf TRANSFORMATION BETWEEN THE CELESTIAL
          AND TERRESTRIAL SYSTEMS}
\bigskip

     The coordinate transformation to be used from the TRS to the
CRS at the date $t$ of the observation can be written as:
$$[{\rm CRS}]=PN(t)R(t)W(t)\ [{\rm TRS}],$$
where $PN(t)$, $R(t)$ and $W(t)$ are the transformation matrices
arising from the motion of the Celestial Ephemeris Pole (CEP) in
the CRS, from the rotation of the Earth around the axis of the
CEP, and from polar motion respectively.

     Two equivalent options can be used.  These two options
have been shown to be consistent within $\pm0.05$ milliseconds of arc
(mas) both theoretically (Capitaine, 1990) and numerically using
existing astrometric data (Capitaine and Chollet, 1991) or
simulated data over two centuries (Capitaine and Gontier, 1991).

     Option 1, corresponding to the classical procedure, makes
use of the equinox for realizing the intermediate reference frame
of date $t$. It uses apparent Greenwich Sidereal Time in the
transformation matrix $R(t)$ and the classical precession and
nutation parameters in the transformation matrix $PN(t)$,

     Option 2 makes use of the Conventional Ephemeris Origin (CEO)
originally referred to as the nonrotating origin (Guinot,
1979) to realize the intermediate reference frame of date $t$:  it
uses the stellar angle (from the origin in the TRS to the CEO in the
CRS) in the transformation matrix $R(t)$ and the two coordinates of
the Celestial Ephemeris Pole in the CRS (Capitaine, 1990) in the
transformation matrix $PN(t)$.  This leads to very simple expressions
of the partial derivatives of observables with respect to
polar coordinates, UT1, and celestial pole offsets.

     The following sections give the details of these two options
as well as the standard expressions necessary to obtain the
numerical values of the relevant parameters at the date of the
observation.  Subroutines for options 1 and 2 of the coordinate
transformation from the TRS to the CRS are available from the
Central Bureau on request together with the development of the
parameters.

     The expressions of the precession and nutation quantities
have been developed originally as functions of barycentric
dynamical time (TDB) defined by IAU recommendations of 1976 and
1979.  In 1991 the IAU adopted definitions of other time scales. 
See Chapter 11 for the relationships among these time scales.  
The parameter $t$ is defined by
$$t=({\rm TT}-2000 {\rm\ January\ 1d\ 12h\ TT)\ in\ days}/36525.$$
This definition is consistent with Resolution C7 passed at the
1994 Hague General Assembly of the IAU which recommends that J2000.0
be defined at the geocenter and at the date 2000.0 January 1.5 TT
= Julian Date 2451545.0 TT.  In the following, $R_1$, $R_2$ and $R_3$ 
denote direct rotations about the axes 1, 2 and 3 of the coordinate 
frame.

\write 0 {\indent Coordinate Transformation Referred to the Equinox 
\noexpand\dotfill\the\pageno\noexpand\break}
\noindent{\bf Coordinate Transformation Referred to the Equinox}

\nobreak

Option 1 uses the form of the coordinate transformation
$$[{\rm CRS}]=PN'(t)R'(t)W'(t)\ [{\rm TRS}],\eqno(1)$$
in which the three fundamental components are (Mueller, 1969)
$$W'(t)=R_1(y_p)\cdot R_2(x_p),$$
$x_p$ and $y_p$ being the ``polar coordinates" of the CEP in the TRS;
$$R'(t)=R_3(-{\rm GST}),$$
GST being Greenwich True Sidereal Time at date $t$, including both
the effect of Earth rotation and the accumulated precession and
nutation in right ascension;  and
$$PN'(t)=[P][N],$$
$${\rm with}\ [P]=R_3(\zeta_A)\cdot R_2(-\theta_A)\cdot R_3(z_A)$$
for the transformation matrix corresponding to the precession
between the reference epoch and the date $t$,
$$[N]=R_1(-\epsilon_A)\cdot R_3(\Delta\psi)\cdot R_1(\epsilon_A+
\Delta\epsilon)$$
for the transformation matrix corresponding to the nutation at
date $t$.

     Standard values of the parameters to be used in form (1) of
the transformation are explained below.

     The standard polar coordinates to be used for the parameters
$x_p$ and $y_p$ (if not estimated from the observations) are those
published by the IERS.  Apparent Greenwich Sidereal Time GST at the 
date t of the observation, must be derived from the following 
expressions:

     (i)  the relationship between Greenwich Mean Sidereal Time
(GMST) and Universal Time as given by Aoki {\it et al.} (1982):
$${\rm GMST}_{\rm 0h\ UT1}=6^h\ 41^m\ 50^s\d 54841 +
8640184\sd 812866T'_u+0\sd 093104T'^2_u-6\sd 2\times 10^{-6}T'^3_u,$$
with $T'_u = d'_u/36525$, $d'_u$ being the number of days elapsed since
2000 January 1, 12h UT1, taking on values $\pm0.5$, $\pm1.5$, ...,

     (ii) the interval of GMST from 0h UT1 to the hour of the
observation in UT1,
$${\rm GMST}={\rm GMST}_{\rm 0h\ UT1}+r[({\rm UT1-UTC)+UTC}],$$
where $r$ is the ratio of universal to sidereal time as given by
Aoki {\it et al.} (1982),
$$r=1.002737909350795+5.9006\times 10^{-11}T'_u-5.9\times 10^{-15}
T'^2_u$$
and the UT1-UTC value to be used (if not estimated from the
observations) is the IERS value.

     (iii)  accumulated precession and nutation in right ascension
(Aoki and Kinoshita, 1983),
$${\rm GST} = {\rm GMST}+\Delta\psi\cos\epsilon_A+0\qd00264\sin\Omega 
+ 0\qd 000063\sin 2\Omega,$$
where $\Omega$ is the mean longitude of the ascending node of the lunar
orbit.  The last two terms have not been included in the IERS
Standards previously.  They should not be included in (iii) until
1 January 1997 when their use will begin.  This date is chosen
to minimize any discontinuity in UT1.  The effect of these terms
on the estimation of UT1 has been described by Capitaine and
Gontier (1993).

     The numerical expression for the precession quantities $\zeta_A$,
$\theta_A$, $z_A$  and $\epsilon_A$ have been given by Lieske
{\it et al.} (1977) as functions of two time parameters $t$ and $T$
(the last parameter representing Julian centuries from J2000.0 to an 
arbitrary epoch). The simplified expressions when the arbitrary epoch
is chosen to be J2000.0 ({\it i.e.} $T = 0$) are
\item{$\zeta_A$} = $2306\qd2181 t + 0\qd30188 t^2 + 0\qd017998 t^3$,
\item{$\theta_A$} = $2004\qd3109 t - 0\qd42665 t^2 - 0\qd041833 t^3$,
\item{$z_A$} = $2306\qd2181 t + 1\qd09468 t^2 + 0\qd018203 t^3$,
\item{$\epsilon_A$} = $84381\qd448 - 46\qd8150 t - 0\qd00059 t^2 +
0\qd001813 t^3$.

     The nutation quantities $\Delta\psi$ and $\Delta\epsilon$ to be
used are the nutation angles in longitude and obliquity.

     For observations requiring values of the nutation angles
with an accuracy of $\pm 1$ mas, it is necessary to add (if those
quantities are not estimated from the observations) the IERS
published values (observed or predicted) for the ``celestial pole
offsets" ({\it i.e.} corrections dpsi and deps).

\write 0 {\indent The IAU 1980 Theory of Nutation \noexpand\dotfill
\the\pageno\noexpand\break}
\noindent{\bf The IAU 1980 Theory of Nutation}

\nobreak

     The IAU 1980 Theory of Nutation (Seidelmann, 1982; Wahr,
1981) is based on a modification of a rigid Earth theory published by
Kinoshita (1977) and on the geophysical model 1066A of
Gilbert and Dziewonski (1975).  It therefore includes the effects
of a solid inner core and a liquid outer core and a ``distribution
of elastic parameters inferred from a large set of seismological
data."  The constants defining the theory are given in Table 5.1.

     VLBI and LLR observations have shown that there are deficiencies
in the IAU 1976 Precession and in the IAU 1980 Theory of Nutation.
However, these models are kept as part of the IERS Standards and the
observed differences ($\delta\Delta\psi$ and $\delta\Delta\epsilon$,
equivalent to dpsi and deps in the IERS Bulletins) with respect to the
conventional celestial pole position defined by the models are
monitored and reported by the IERS as ``celestial pole offsets." 
Using these offsets the corrected nutation is given by
$$\Delta\psi=\Delta\psi({\rm IAU\ 1980})+\delta\Delta\psi,$$
and
$$\Delta\epsilon=\Delta\epsilon({\rm IAU 1980})+\delta\Delta\epsilon.$$
This is practically equivalent to replacing $N$ with the rotation
described by Lieske (1991),
$$\tilde{N}={\cal R}N_{\rm IAU}.$$
where 
$${\cal R}=\pmatrix{1&-\delta\Delta\psi\cos\epsilon_t&
-\delta\Delta\psi\sin\epsilon_t\cr\delta\Delta\psi\cos\epsilon_t&
1&\delta\Delta\epsilon\cr\delta\Delta\psi\sin\epsilon_t&
\delta\Delta\epsilon&1\cr},$$
$\epsilon_t=\epsilon_A+\Delta\epsilon$, 
and $N_{\rm IAU}$ represents the IAU 1980 Theory of Nutation according 
to which 
$$\Delta\psi=\sum_{i=1}^{106}(A_i+A'_it)\sin({\rm ARGUMENT}),
\kern1pc
\Delta\epsilon=\sum_{i=1}^{106}(B_i+B'_it)\cos({\rm ARGUMENT}).$$
Here 
$${\rm ARGUMENT} = \sum N_i F_i,$$ 
the $N_i, (i=1,\cdots,5)$ being integers multiplying the Fundamental 
Arguments $F_i$ of nutation theory, namely,  
\medskip
\halign{\kern1pc#\hfil&\kern1em#\hfil\cr\cr
$F_1\equiv l$&  = Mean Anomaly of the Moon\cr
&$= 134\dd96340251 + 1717915923\qd2178 t 
    + 31\qd8792 t^2 + 0\qd051635 t^3 - 0\qd00024470 t^4$,\cr
\cr
$F_2\equiv l'$& = Mean Anomaly of the Sun\cr
&$= 357\dd52910918 + 129596581\qd0481 t
    - 0\qd5532 t^2 + 0\qd000136 t^3 - 0\qd00001149 t^4$,\cr
\cr
$F_3\equiv F$&  = $L - \Omega$\cr
&$= 93\dd27209062 + 1739527262\qd8478 t
     - 12\qd7512 t^2 - 0\qd001037 t^3 + 0\qd00000417 t^4$,\cr
\cr
$F_4\equiv D$&  = Mean Elongation of the Moon from the Sun\cr
&$= 297\dd85019547 + 1602961601\qd2090 t
     - 6\qd3706 t^2 + 0\qd006593 t^3 - 0\qd00003169 t^4$,\cr
\cr
$F_5\equiv\Omega$&  = Mean Longitude of the Ascending Node of the Moon\cr
&$= 125\dd04455501 - 6962890\qd2665 t
     + 7\qd4722 t^2 + 0\qd007702 t^3 - 0\qd00005939 t^4$,\cr
\cr}
\noindent where $t$ is measured in Julian Centuries of 36525 days of
86400 seconds of Dynamical Time since J2000.0 and where $1^r =
360^\circ = 1296000\qd0$ (Simon {\it et al.}, 1994). $L$ is the Mean 
Longitude of the Moon.

The reader is cautioned that there may be an ambiguity in the 
assignment of the set of five multipliers $N_j$ to any particular 
term of the nutation series. This ambiguity is illustrated, and a 
convention for unique assignment of multipliers for prograde and 
retrograde circular nutations is presented, in a note at the end of 
this chapter.

\bigskip
\noindent Table 5.1.  IAU 1980 Theory of Nutation in longitude 
$\Delta\psi$ and obliquity $\Delta\epsilon$, referred to the mean 
equator and equinox of date, with $t$ measured in Julian centuries 
from epoch J2000.0.  The signs of the fundamental arguments, periods,
and coefficients may differ from the original publication.  These
have been changed to be consistent with other portions of this 
chapter.
$$\Delta\psi=\sum_{i=1}^{106}(A_i+A'_it)\sin({\rm ARGUMENT}),
\kern1pc
\Delta\epsilon=\sum_{i=1}^{106}(B_i+B'_it)\cos({\rm ARGUMENT}).$$
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 2&  0&  0&  2&  0&     7.1&        1&     0.0&      0&    0.0\cr
 2&  0&  2& -2&  1&    12.8&        1&     0.0&     -1&    0.0\cr
 2&  0& -2&  0&  1&  -943.2&        1&     0.0&      0&    0.0\cr
 1& -1&  0& -2&  0&   -29.3&        1&     0.0&      0&    0.0\cr
-1&  0&  0&  1&  1&  -388.3&        1&     0.0&      0&    0.0\cr
-1& -1&  0&  2&  1&    35.0&        1&     0.0&      0&    0.0\cr
 0&  1&  0&  1&  0&    27.3&        1&     0.0&      0&    0.0\cr}
$$\epsilon_0=23^\circ 26' 21\qd448$$
$$\sin\epsilon_0=0.39777716$$
}
\bigskip
\hrule
\bigskip

\write 0 {\indent The IERS 1996 Theory of Precession/Nutation \noexpand\dotfill
\the\pageno\noexpand\break}
\noindent{\bf The IERS 1996 Theory of Precession/Nutation}

\nobreak

     At the request of the IERS, T. Herring (1996) has analyzed
the most recent VLBI and LLR data for geophysical parameters
required to adjust the rigid Earth theory of nutation to the non-
rigid theory.  The non-rigid Earth theory coefficients resulting
from this procedure are shown in Table 5.2.  Table 5.3 contains 
the planetary nutation terms and the necessary planetary
arguments.  These coefficients are meant to be used for
prediction purposes and by those requiring accurate {\it a priori}
estimates of nutation.  They are not meant to replace the IAU 
Theory.  The IERS will continue to publish observed estimates of
the corrections to the IAU 1976 Precession and the IAU 1980 Theory 
of Nutation in its publications. 

The use of the IERS 1996 Theory of nutation must also be associated
with the use of improved numerical values for the precession rate
of the equator in longitude and obliquity:
$$\delta\psi_A=-0.2957''/{\rm c}\kern.2cm{\rm and}\kern.2cm\delta\omega_A=-0.0227''/{\rm c}$$

\noindent Table 5.2.  IERS 1996 series for nutation in longitude 
$\Delta\psi$ and obliquity $\Delta\epsilon$, referred to the mean 
equator and equinox of date, with $t$ measured in Julian centuries 
from epoch J2000.0.  The signs of the fundamental arguments, periods,
and coefficients may differ from the original publication.  These
have been changed to be consistent with other portions of this 
chapter.
$$\Delta\psi=\sum_{i=1}^{263}(A_i+A'_it)\sin({\rm ARGUMENT})
+A''_i\cos({\rm ARGUMENT}),$$
$$\Delta\epsilon=\sum_{i=1}^{263}(B_i+B'_it)\cos({\rm ARGUMENT})
+B''_i\sin({\rm ARGUMENT}).$$

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-1&-1& 2& 0& 2&   29.26&       -16&      0&       6&    0&    0&    0\cr
-2&-1& 0& 2& 1& -129.17&       -13&      0&       7&    0&    0&    0\cr
 0& 0& 0&-2& 2&  -14.70&        13&      0&      -5&    0&    0&    0\cr
 0&-2& 2& 2& 2&    7.38&       -13&      0&       5&    0&    0&    0\cr
 1& 0& 0&-4& 1&  -10.07&       -12&      0&       6&    0&    0&    0\cr
-1& 1& 0& 2& 1&   29.39&       -10&      0&       6&    0&    0&    0\cr
-2& 0& 0& 4& 1&   15.94&        11&      0&      -6&    0&    0&    0\cr
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-1& 0&-2& 0& 1&   -9.10&       -10&      0&       5&    0&    0&    0\cr
-1& 1& 2& 0& 1&   25.13&         9&      0&      -5&    0&    0&    0\cr
-1& 1& 2&-2& 1&  -35.80&       -11&      0&       5&    0&    0&    0\cr
-1&-1& 0& 4& 0&   10.37&        15&      0&       0&    0&    0&    0\cr
-3& 0& 0& 4& 0&   37.63&        16&      0&       0&    0&    0&    0\cr
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 0& 2&-2& 2& 0&-3396.16&        -9&      0&       0&    0&    0&    0\cr
 0&-1& 2& 4& 2&    4.86&        -9&      0&       0&    0&    0&    0\cr
 0&-1& 2&-1& 2&   27.32&         9&      0&       0&    0&    0&    0\cr
 1& 1& 0& 2& 0&    9.37&       -10&      0&       0&    0&    0&    0\cr
 2& 0& 0&-2& 2&  219.17&       -11&      0&       0&    0&    0&    0\cr
 2&-1& 2& 2& 2&    4.74&        -9&      0&       0&    0&    0&    0\cr
 4& 0& 0& 0& 0&    6.89&         9&      0&       0&    0&    0&    0\cr
 4& 0& 2&-2& 2&    6.64&        12&      0&       0&    0&    0&    0\cr
-1& 0& 0& 3& 0&   15.31&       -10&      0&       0&    0&    0&    0\cr
-1& 0& 4&-2& 2&   23.43&        -9&      0&       0&    0&    0&    0\cr
-1&-2& 2& 2& 2&   10.08&        -9&      0&       0&    0&    0&    0\cr
-2&-1& 0& 4& 0&   16.63&        12&      0&       0&    0&    0&    0\cr
-2&-1& 2& 4& 2&    7.50&       -12&      0&       0&    0&    0&    0\cr
 0& 1& 2& 2& 1&    6.95&         7&      0&       0&    0&    0&    0\cr
 0& 2& 2& 0& 2&   12.71&         7&      0&       0&    0&    0&    0\cr
 0&-2& 2& 0& 2&   14.77&        -8&      0&       0&    0&    0&    0\cr
 1& 0& 0& 4& 0&    5.82&         8&      0&       0&    0&    0&    0\cr
 1& 0& 2& 2& 0&    5.63&         8&      0&       0&    0&    0&    0\cr
 1& 0& 2&-4& 2&  -38.52&         7&      0&       0&    0&    0&    0\cr
 1&-1& 2& 2& 1&    5.73&        -8&      0&       0&    0&    0&    0\cr
 1&-1& 2&-2& 2&   25.62&        -7&      0&       0&    0&    0&    0\cr
 1&-2& 0& 0& 0&   32.45&         8&      0&       0&    0&    0&    0\cr
 2& 0& 0& 0& 2&   13.83&        -8&      0&       0&    0&    0&    0\cr
 2& 1& 0&-2& 1&  134.27&         8&      0&       0&    0&    0&    0\cr
 3& 0& 0& 0& 1&    9.20&         7&      0&       0&    0&    0&    0\cr
-1& 0& 2& 1& 2&   14.13&         8&      0&       0&    0&    0&    0\cr
-1& 0& 2& 3& 2&    7.22&         8&      0&       0&    0&    0&    0\cr
-1& 0&-2& 4& 0&   38.96&        -7&      0&       0&    0&    0&    0\cr
-1& 1& 2& 2& 1&    9.30&         7&      0&       0&    0&    0&    0\cr
-1& 2& 0& 2& 0&   27.09&        -8&      0&       0&    0&    0&    0\cr
-1&-1& 2&-1& 1& 2189.73&         7&      0&       0&    0&    0&    0\cr
-2& 0& 2&-2& 1&  -14.93&        -8&      0&       0&    0&    0&    0\cr
-2& 0& 4& 0& 2&   13.49&        -7&      0&       0&    0&    0&    0\cr
-2& 0&-2& 2& 0&  -12.76&         8&      0&       0&    0&    0&    0\cr
-2& 1& 2& 0& 1&  285.41&         9&      0&       0&    0&    0&    0\cr
-3& 0& 2& 0& 1&  -28.15&        -8&      0&       0&    0&    0&    0\cr
 0& 1& 0& 1& 1&   27.43&         5&      0&       0&    0&    0&    0\cr
 0&-1& 0& 4& 0&    7.53&         6&      0&       0&    0&    0&    0\cr
 0&-1& 0&-2& 1&  -14.16&         5&      0&       0&    0&    0&    0\cr
 0&-2& 0& 0& 1& -177.85&        -6&      0&       0&    0&    0&    0\cr
 1& 0& 2& 1& 1&    6.97&         5&      0&       0&    0&    0&    0\cr
 1& 0& 2&-3& 2&  126.51&        -6&      0&       0&    0&    0&    0\cr
 1& 0&-2& 1& 0& -299.26&        -7&      0&       0&    0&    0&    0\cr
 1& 1& 0& 1& 0&   13.72&         5&      0&       0&    0&    0&    0\cr
 1&-1& 0&-2& 1&  -29.14&         6&      0&       0&    0&    0&    0\cr
 2& 0& 2&-1& 2&    8.93&        -6&      0&       0&    0&    0&    0\cr
 2& 1& 2& 0& 1&    6.73&         5&      0&       0&    0&    0&    0\cr
 2&-1& 2& 0& 1&    6.98&        -6&      0&       0&    0&    0&    0\cr
 2&-1& 2&-2& 2&   13.28&         5&      0&       0&    0&    0&    0\cr
 3& 0& 0& 2& 0&    5.66&         5&      0&       0&    0&    0&    0\cr
 3&-1& 2& 0& 2&    5.58&        -5&      0&       0&    0&    0&    0\cr
-1&-1& 2& 0& 1&   29.14&        -6&      0&       0&    0&    0&    0\cr
-2& 0& 0& 0& 2&  -13.72&         6&      0&       0&    0&    0&    0\cr
-2& 0& 0& 3& 0&   34.48&        -5&      0&       0&    0&    0&    0\cr
-2& 0& 0&-2& 1&   -7.12&        -5&      0&       0&    0&    0&    0\cr
-2& 0& 2& 2& 0&   14.57&        -6&      0&       0&    0&    0&    0\cr
-2&-1& 2& 0& 0& -548.04&        -5&      0&       0&    0&    0&    0\cr
-2&-1& 2& 2& 2&   15.24&         6&      0&       0&    0&    0&    0\cr
 0& 0& 1& 0& 0&   27.21&         0&      0&       0&    0&    8&    0\cr
 0& 0& 1& 0& 1&   27.32&         0&      0&       0&    0&  -16&  -14\cr
-1& 0& 1& 0& 0& 2190.35&         0&      0&       0&    0&   33&    0\cr
-1& 0& 1& 0& 1& 3231.51&         0&      0&       0&    0& -105&  -89\cr
-1& 0& 1& 0& 2& 6159.22&         0&      0&       0&    0&   36&   18\cr
-1& 0& 1& 0& 3&65514.10&         0&      0&       0&    0&   -6&    0\cr}}
\bigskip
\hrule

\noindent Table 5.3.  IERS 1996 Series for planetary nutation in 
longitude $\Delta\psi$ and obliquity $\Delta\epsilon$, referred to the 
mean equator and equinox of date, with $t$ measured in Julian centuries
from epoch J2000.0.  The signs of the fundamental arguments, periods,
and coefficients may differ from the original publication.  These
have been changed to be consistent with other portions of this
chapter.
$$\Delta\psi=\sum_{i=1}^{112}A_i\sin({\rm ARGUMENT})+
A_i'\cos({\rm ARGUMENT}),$$
%\kern1pc
$$\Delta\epsilon=\sum_{i=1}^{112}B_i\sin({\rm ARGUMENT})+
B_i'\cos({\rm ARGUMENT}).$$

{\small
\baselineskip = 0.7 pc
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MULTIPLIERS OF& PERIOD& LONGITUDE& OBLIQUITY\cr 
&&$(0.001 mas)$&  $(0.001 mas)$\cr}
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$l_{Ve}$&$l_E$&$l_{Ma}$&$l_J$&$l_{Sa}$&$p_a$&$D$&$F$&$l$&$\Omega$&(days)&
$A_i$&$A'_i$&$B_i$&$B'_i$\cr
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  0&  2&  0& -3&  0&  0&  0&  0&  0&  0&   209.07&   8&   2&   0&   0\cr
  2& -5&  0&  0&  0& -2&  0&  0&  0&  0&  -208.83&   0& -13&   6&   0\cr
  6& -8&  0&  0&  0& -2&  0&  0&  0&  0&   208.35& -12&   0&   0&  -5\cr
  0&  2&  0& -2&  0&  0&  0&  0&  0&  0&   199.44&  39&   0&   0&   0\cr
  3& -3&  0&  0&  0&  0&  0&  0&  0&  0&   194.64&  10&   0&   0&   0\cr
  3& -3&  0&  0&  0&  2&  0&  0&  0&  0&   194.63&   5&  -2&   0&   0\cr
  0& -2&  0&  1&  0& -2&  0&  0&  0&  0&  -190.66&  15&   3&  -1&   7\cr
  0&  3& -2&  0&  0&  2&  0&  0&  0&  0&   188.60&   8&  -7&   0&   0\cr
  8&-15&  0&  0&  0& -2&  0&  0&  0&  0&  -183.00&  -6& -10&   4&  -3\cr
  0&  6& -8&  3&  0&  2&  0&  0&  0&  0&   182.67&  12& -42& -18&  -5\cr
  0&  2&  0&  0&  0&  2&  0&  0&  0&  0&   182.62&  -9&   0&   0&   0\cr
  0& -2&  8& -3&  0&  2&  0&  0&  0&  0&   182.57& -12&  42&  18&   5\cr
 -8& 11&  0&  0&  0& -2&  0&  0&  0&  0&  -182.24&   6&  10&  -4&  13\cr
  0&  1&  2&  0&  0&  2&  0&  0&  0&  0&   177.01&  -8&  -7&   0&   0\cr
  0&  2&  0&  1&  0&  2&  0&  0&  0&  0&   175.23&  17&  -1&   0&  -7\cr
  3& -7&  0&  0&  0& -2&  0&  0&  0&  0&  -172.01&   7&  -2&   0&   0\cr
  2& -1&  0&  0&  0&  2&  0&  0&  0&  0&   162.26&   0& -17&  -8&   0\cr
  7& -9&  0&  0&  0& -2&  0&  0&  0&  0&   153.56&  -7&   0&   0&   0\cr
  4& -4&  0&  0&  0&  0&  0&  0&  0&  0&   145.98&  11&   0&   0&   0\cr
  1&  1&  0&  0&  0&  2&  0&  0&  0&  0&   139.11& -30&   0&   0&  13\cr
  0&  3&  0& -2&  0&  2&  0&  0&  0&  0&   129.00&   7&  -9&  -4&  -3\cr
  3& -2&  0&  0&  0&  2&  0&  0&  0&  0&   126.97&   0& -11&  -5&   0\cr
  0&  3&  0& -1&  0&  2&  0&  0&  0&  0&   125.27&  52&   2&   0& -22\cr
  0&  4& -2&  0&  0&  2&  0&  0&  0&  0&   124.38&  14&   0&   0&  -6\cr
  8&-10&  0&  0&  0& -2&  0&  0&  0&  0&   121.59&  -5&   0&   0&   0\cr
  5& -5&  0&  0&  0&  0&  0&  0&  0&  0&   116.78&   7&   0&   0&   0\cr
  2&  0&  0&  0&  0&  2&  0&  0&  0&  0&   112.35&  39&   0&   0& -17\cr
  0&  4&  0& -2&  0&  2&  0&  0&  0&  0&    95.33& -18&   0&   0&   8\cr
-18& 16&  0&  0&  0&  0&  0& -2&  1& -2&   -13.66&  13&   6&  -3&   5\cr
-18& 16&  0&  0&  0&  0&  0&  2&  1&  2&    13.66&  13&  -6&  -3&  -5\cr}}
$$l_{Ve} = 181\dd979800853 + 58517\dd8156748\times t$$
$$l_{E} = 100\dd466448494 + 35999\dd3728521\times t$$
$$l_{Ma} = 355\dd433274605 + 19140\dd299314\times t$$
$$l_{Ju} = 34\dd351483900 + 3034\dd90567464\times t$$
$$l_{Sa} = 50\dd0774713998 + 1222\dd11379404\times t$$
$$p_{a} = 1\dd39697137214\times t + 0\dd0003086\times t^2$$
\hrule

\write 0 {\indent The Multipliers of the Fundamental Arguments of
Nutation Theory \noexpand\dotfill\the\pageno\noexpand\break}
\noindent {\bf The multipliers of the fundamental arguments of nutation
theory}

\nobreak

Each of the lunisolar terms in the nutation series is characterized by 
a set of five integers $N_j$ which determines the ARGUMENT for the term 
as a linear combination of the five Fundamental Arguments 
$F_j$, namely the Delaunay variables $(\ell, \ell', F, D, \Omega)$: 
${\rm ARGUMENT} = \sum_{j = 1}^5 N_j F_j \equiv \vec{N}\cdot \vec{F},$
where $\vec{N}$ is the five-vector composed of the values 
($N_1,\cdots ,N_5$) which characterize the term, and $\vec{F}$ is the 
five-vector ($F_1,\cdots,F_5$). The $F_j$ are functions of time, and 
the angular frequency of the nutation described by the term is given by 

$$\omega \equiv {d({\rm ARGUMENT})\over dt}.$$ 

\noindent The frequency thus defined is positive for most terms, and 
negative for some. Planetary nutation terms differ from the above only 
in that ${\rm ARGUMENT} = \sum_{j=1}^{10} N'_j F'_j$ as noted in Table 
5.3.

Over time scales involved in nutation studies, the frequency $\omega$ 
is effectively time-independent, and one may write, for the $s$th term 
in the nutation series, 
$${\rm ARGUMENT} = \omega_s t +\alpha_s.$$

Different tables of nutations in longitude and obliquity do not  
necessarily assign the same set of multipliers $N_j$ to a particular 
term in the nutation series. Compare, for instance, the 31.8 day 
term in Table 5.1 wherein $\vec {N} = (-1,0,0,2,0)$ with the same term 
in Seidelmann (1982) where the assignment is $\vec {N} = (1,0,0,-2,0)$. 
The differences in the assignments arises from the fact that the 
replacement $\vec {N}_j \to -\vec {N}_j$ accompanied by reversal of 
the sign of the coefficient of $\sin({\rm ARGUMENT}_j)$ in the series 
for $\Delta \psi$ and $\Delta \epsilon$ leaves these series unchanged. 
This freedom has led to some confusion in the assignment of multipliers 
of the Delaunay arguments for amplitudes of retrograde and prograde 
circular nutations. 

The nutations in longitude and obliquity at a given frequency 
$\omega_s$ may be viewed as the superposition of a pair of prograde 
and retrograde circular nutations whose frequencies have the same 
absolute value, $|\omega_s|$, but with opposite signs. These circular 
nutation terms may be written as 
$$A_s^{pro} e^{-iq_s(\omega_s t + \alpha_s)}\quad {\rm and} \quad 
A_s^{ret} e^{ iq_s(\omega_s t + \alpha_s)},$$
where $q_s$ is the sign of $\omega_s$: 
$$q_s \omega_s = |\omega_s|.$$
With the convention that the exponential factor involving the frequency
$\nu$ is to be written as $e^{-i\nu t}$, one sees from the above 
expressions that the frequency of the prograde nutation, $\omega_s^
{pro}=(q_s\omega_s) \equiv |\omega_s|$, is positive, and that of the 
retrograde nutation is negative ($\omega_s^{ret}=-|\omega_s|$). 

Now, given the vector $\vec {N}$ of multipliers of the fundamental 
arguments for the nutations in longitude and obliquity of frequency 
$\omega_s$, one can choose multipliers $\vec {N}^{(pro)}$ and 
$\vec {N}^{(ret)}$ for the corresponding prograde and retrograde 
nutations in such a manner as to ensure that 
%
$$\omega_s^{pro} \equiv \vec{N}^{pro}\cdot {d \vec{F}\over dt} = 
|\omega_s|, \quad \omega_s^{ret}  \equiv \vec{N}^{ret}\cdot {d \vec{F}
\over dt} = -|\omega_s|.$$
%
This is accomplished by making the assignments 
%
$$\vec{N}^{pro} = q_s \vec{N}; \quad \vec{N}^{ret} = -q_s \vec{N},$$
%
\noindent a practice that will be adhered to in this work.  Note that 
$\vec {N}^{(pro)}$ and $\vec {N}^{(ret)}$ are unaffected by whether 
$\vec {N}$ or $-\vec {N}$ is used to label the nutations in longitude 
and obliquity, because on switching from one to the other, the sign of 
$\omega_s$ also is switched simultaneously. 

\write 0 {\indent Conversion to Prograde and Retrograde Nutation 
Amplitudes \noexpand\dotfill\the\pageno\noexpand\break}
\noindent{\bf Conversion to Prograde and Retrograde Nutation Amplitudes}
\nobreak

With the convention for the time dependence of the circular nutation
amplitudes as stated above, the amplitudes of prograde and retrograde 
circular nutations are written as 
$$A_s^{pro}=A_s^{pro \; ip} - iA_s^{pro \; op},$$
%
$$A_s^{ret}=A_s^{ret \; ip} - iA_s^{ret \; op},$$
% 
\noindent where $A_s^{pro \; ip}$ and $A_s^{pro \; op}$ are the 
in-phase and out-of-phase parts of the prograde nutation with angular
frequency $|\omega_s|$ as defined in the foregoing section, 
and $A_s^{ret \; ip}$ and $A_s^{ret \; op}$ are the in-phase and 
out-of-phase parts of the retrograde nutation with 
angular frequency $(-|\omega_s|)$.
These amplitudes are related to the coefficients in longitude and 
obliquity through the following equations (see Defraigne {\it et al.,} 
1995): 
%
$$A_s^{pro \; ip} = -{{1}\over{2}} \left( \Delta \epsilon_s^{ip} 
- q_s \Delta \psi_s^{ip} \sin \epsilon_0 \right),$$
$$A_s^{ret \; ip} = -{{1}\over{2}} \left( \Delta \epsilon_s ^{ip} 
+ q_s \Delta \psi_s^{ip} \sin \epsilon_0 \right),$$
$$A_s^{pro \; op} = {{1}\over{2}} \left(q_s \Delta \epsilon_s^{op} 
+  \Delta \psi_s^{op} \sin \epsilon_0 \right),$$
$$A_s^{ret \; op} = -{{1}\over{2}} \left(q_s \Delta \epsilon_s^{op} 
- \Delta \psi_s^{op} \sin \epsilon_0 
\right).$$

\noindent $\Delta \epsilon_s ^{ip}$ and $\Delta \epsilon_s ^{op}$ are
the in-phase (cosine) and out-of-phase (sine) parts of the nutation in 
obliquity for a nutation of frequency $\omega_s$; $\Delta \psi_s^{ip}$ 
and $\Delta \psi_s^{op}$ are the in-phase (sine) and out-of-phase 
(cosine) parts of the nutation in longitude for the same nutation. 

The relations stated above are valid when the convention regarding the 
time dependence as stated in the previous section is followed, {\it 
i.e.}, when the time dependence of the prograde amplitude is $e^{-i|
\omega_s| t}$ and that of the retrograde amplitude is $e^{i|\omega_s|
t}$. However the opposite convention is often used in the literature. 
The amplitudes to be employed in such a case are the complex conjugates 
of those given above, {\it i.e.}, $(A_s^{pro \; ip} + i A_s^{pro \; 
op})$ goes together with $e^{i|\omega_s| t}$ for the prograde nutation 
and $(A_s^{ret \; ip} + iA_s^{ret \; op})$ with $e^{-i|\omega_s| t}$ 
for the retrograde nutation. Since the real parts are to be eventually 
taken, switching between mutually complex conjugates versions does not 
affect the results. 

\write 0 {\indent Coordinate Transformation Referred to the Nonrotating 
Origin \noexpand\dotfill\the\pageno\noexpand\break}
\noindent{\bf Coordinate Transformation Referred to the Nonrotating
Origin}

\nobreak

Option 2 uses form (2) of the coordinate transformation from the
TRS to the CRS
$$[{\rm CRS}]=PN''(t)R''(t)W''(t)[{\rm TRS}],\eqno(2)$$
where the three fundamental components of (2) are given below
(Capitaine, 1990)
$$W''(t)=R_3(-s')\cdot R_1(y_p)\cdot R_2(x_p),$$
$x_p$ and $y_p$ being the ``polar coordinates" of the CEP in the TRS and
$s'$ the accumulated displacement of the terrestrial origin on the true
equator due to polar motion.  The use of the quantity $s'$ (which is
neglected in the classical form (1)) provides an exact realization
of the ``instantaneous prime meridian."
$$R''(t)=R_3(-\theta),$$
$\theta$ being the stellar angle at date t due to the Earth's angle of
rotation,
$$PN''(t)=R_3(-E)\cdot R_2(-d)\cdot R_3(E)\cdot R_3(S),$$
$E$ and $d$ being such that the coordinates of the CEP in the CRS are
$X = \sin d \cos E$, $Y = \sin d \sin E$, $Z = \cos d$ and $S$ being the
accumulated rotation (between the epoch and the date $t$) of the
celestial CEO on the true equator due to the celestial motion of
the CEP. \ $PN''(t)$ can be given in an equivalent form involving
directly $X$ and $Y$ (to which all the observations of a celestial
object from the Earth are actually sensitive) as:
$$PN''(t)=Q_1=\pmatrix{1-aX^2& -aXY& X\cr -aXY& 1-aY^2& Y\cr -X& -Y&
1-a(X^2+Y^2)\cr}\cdot R_3(s),$$
with $a = 1/(1+\cos d)$, which can also be written, with sufficient
accuracy as $a = 1/2 + 1/8 (X^2+Y^2)$.

     The standard values of the parameters to be used in the form
(2) of the transformation are detailed below.

     The standard pole coordinates to be used for the parameters $x_p$
and $y_p$ (if not estimated from the observations) are those published
by the IERS.  The quantity $s'$ (of the order of 0.1 mas/cy) is:  
$$s'= 0.0015(a_c^2/1.2 + a_a^2) t,$$
$a_c$ and $a_a$ being the average amplitudes (in arc seconds) of the
Chandlerian and annual wobbles, respectively in the period
considered (Capitaine {\it et al.}, 1986).

     The stellar angle is obtained by the use of the conventional
relationship between the stellar angle $\theta$, the hour angle of the
CEO and UT1 as given by Capitaine {\it et al.}, (1986),
$$\theta(T_u)=2\pi(0.779057273264+1.00273781191135448T_u\times 36525),$$
where $T_u$ = (Julian UT1 date - 2451545.0)/36525, and

     UT1 = UTC + (UT1-UTC), or equivalently  
$$\theta(T_u)=2\pi({\rm UT1\ Julian\ day\ number\ elapsed\ since\ }
2451545.0 + 0.779057273264$$
$$+ 0.00273781191135448 T_u \times 36525),$$
the quantity UT1-UTC to be used (if not estimated from the
observations) being the IERS value.

     The celestial coordinates $X$ and $Y$ of the CEP to be used for 
consistency with the IAU 1980 theory of nutation are the standard
values as derived from the series given in Table 5.4.  These 
developments of the celestial polar coordinates have been 
derived (Capitaine, 1990) from the previous standard
expressions for precession and nutation with a consistency of $5\times
{10^{-5}}$ sec. of arc after a century; such consistency has been
numerically checked over two centuries (Gontier, 1990). For
observations requiring values of the nutation angles with a
milliarcsecond accuracy, it is necessary to add (if those quantities
are not estimated from the observations) the IERS published values
(observed or predicted) for the ``celestial pole offsets" ({\it i.e.}
corrections $dX = \Delta \psi \sin\epsilon_0$ and $dY = \Delta 
\epsilon$).

     For consistency with the IERS 1996 series for nutation as given
by Tables 5.2 and 5.3, it is necessary to use the following expressions
for $X$ and $Y$ (Capitaine, 1990):
$$X = \sin\omega\sin\psi,$$
$$Y = -\sin\epsilon_0\cos\omega + \cos\epsilon_0\sin\omega\cos\psi,$$
where $\epsilon_0$ is the obliquity of the ecliptic at J2000, $\omega$
is the inclination of the true equator of date on the fixed ecliptic 
of epoch and $\psi$ is the longitude, on the ecliptic of epoch of the
node of the true equator of date on the fixed ecliptic of epoch; these
quantities are such that:
$$\omega = \omega_A + \Delta\epsilon_1 , \psi = \psi_A +\Delta\psi_1,$$
where $\psi_A$ and $\omega_A$ are the precession quantities in longitude
and obliquity (Lieske {\it et al.} 1977) referred to the ecliptic of 
epoch and $\Delta\psi_1$, $\Delta\epsilon_1$ are the nutation angles in
longitude and obliquity referred to the ecliptic of epoch. $\Delta\psi_1$,
$\Delta\epsilon_1$ can be obtained with an accuracy better than one
microarcsecond after one century from the nutation angles $\Delta\psi$,
$\Delta\epsilon$ in longitude and obliquity referred to the ecliptic of 
date, following Aoki and Kinoshita (1983) by:
$$\Delta\psi_1 = {(\Delta\psi\sin\epsilon_A\cos\chi_A - 
\Delta\epsilon\sin\chi_A)\over \sin\omega_A},$$
$$\Delta\epsilon_1=\Delta\psi\sin\epsilon_A\sin\chi_A+
\Delta\epsilon\cos\chi_A,$$
$\omega_A$ and $\epsilon_A$ being the precession quantities in obliquity 
referred to ecliptic of epoch and ecliptic of date respectively and
$\chi_A$ the precession quantity for planetary precession along the
equator (Lieske {\it et al.} 1977).

     The standard value of s compatible with the IAU 1980 Theory of
Nutation and the Lieske {\it et al.} (1977) precession can be derived 
with an accuracy of $5\times {10^{-5}}''$ after a century (Capitaine, 
1990) from the following numerical development and the numerical values 
of $X$ and $Y$ (Table 5.3),
$$s=-XY/2+0\qd00385t-0\qd07259t^3-0\qd00264\sin\Omega-
0\qd00006\sin2\Omega$$
$$+0\qd00074t^2\sin\Omega+0\qd00006t^2\sin 2(F-D+\Omega)$$

\vfil\eject
\bigskip
\noindent Table 5.4  Series for the celestial coordinates $X$ and $Y$
of the CEP referred to the mean equator and equinox of epoch J2000.0,
with $t$ measured in Julian centuries from epoch J2000.0. The terms
following the dotted line are identical in Tables 5.1 and 5.4.  The 
signs of the fundamental arguments, periods, and coefficients may 
differ from the original publication.  These have been changed to be 
consistent with other portions of this chapter.
\smallskip
\hrule
\medskip
\halign{#&#\hfil\cr
$X$&$=2004\qd3109t - 0\qd42665t^2 - 0\qd198656t^3 + 0\qd0000140t^4$\cr
&$+0\qd00006t^2\cos\Omega+\sin\epsilon_0\{\sum[(A_i+A'_it)
\sin({\rm ARGUMENT})+A''_it\cos({\rm ARGUMENT})]\}$\cr
&$ + 0\qd00204t^2\sin\Omega+0\qd00016t^2\sin 2(F-D+\Omega),$\cr
\cr
$Y$&$ = - 0\qd00013-22\qd40992t^2+0\qd001836t^3+0\qd0011130t^4$\cr
&$ + \sum[(B_i+B'_it)\cos({\rm ARGUMENT})+B''_it
\sin({\rm ARGUMENT})]$\cr 
&$ - 0\qd00231t^2\cos\Omega-0\qd00014t^2\cos2(F-D+\Omega)$\cr}
{\small
\baselineskip = 0.7 pc
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$\big(0\qd0001\big)$ \kern0.5pc OBLIQUITY $\big(0\qd0001\big)$
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 0& 0& 0& 2&0&   14.8&     63&  &  0.0&   0.0&   -2&  & 0.0&  -0.6\cr
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 1& 0& 2& 0&1&    9.1&    -51&  &  0.0&   1.5&   27&  & 0.0&   0.5\cr
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-2& 0& 2& 0&1& 1305.5&     46&  &  0.0&  -1.3&  -24&  & 0.0&   0.0\cr
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 0& 0& 2& 2&2&    7.1&    -38&  &  0.0&      &   16&  & 0.0&\cr
 2& 0& 2& 0&2&    6.9&    -31&  &  0.0&      &   13&  & 0.0&\cr
 2& 0& 0& 0&0&   13.8&     29&  &  0.0&      &   -1&  & 0.0&\cr
 1& 0& 2&-2&2&   23.9&     29&  &  0.0&      &  -12&  & 0.0&\cr
 0& 0& 2& 0&0&   13.6&     26&  &  0.0&      &   -1&  & 0.0&\cr
 0& 0& 2&-2&0&  173.3&    -22&  &  0.0&      &    0&  & 0.0&\cr
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 0& 2& 0& 0&0&  182.6&     17&  & -0.1&      &    0&  & 0.0&\cr
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 1& 0& 0& 2&1&   32.0&     16&  &  0.0&      &   -8&  & 0.0&\cr
 0& 1& 0& 0&1&  386.0&    -15&  &  0.0&      &    9&  & 0.0&\cr
 1& 0& 0&-2&1&  -31.7&    -13&  &  0.0&      &    7&  & 0.0&\cr
 0&-1& 0& 0&1& -346.6&    -12&  &  0.0&      &    6&  & 0.0&\cr
 2& 0&-2& 0&0&-1095.2&     11&  &  0.0&      &    0&  & 0.0&\cr
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\bigskip
\hrule


\write 0 {\indent Geodesic Nutation \noexpand\dotfill\the\pageno
\noexpand\break}
\noindent{\bf Geodesic Nutation}

\nobreak

     Fukushima (1991) has pointed out that, if extreme precision is
required, the effect of geodesic nutation must be taken into
account.  For Option (1) this would require a correction in
longitude of
$$\Delta\psi_g = -0\qd000153\sin l' - 0\qd000002\sin 2l',$$
where $l'$ is the mean anomaly of the Sun.  For Option (2) it would
require a correction to $X$ of
$$\Delta X_g=(-0\qd0000609\sin l'-0\qd0000008\sin 2l')\sin\epsilon_0.$$
In both cases, the correction would be added to the uncorrected
determination of $\psi$ or $X$. 

\noindent{\bf References}

\nobreak

\item{}\kern-2pc
Aoki, S., Guinot, B., Kaplan, G. H., Kinoshita, H., McCarthy, D.
     D., and Seidelmann, P. K., 1982, ``The New Definition of Universal
     Time," {\it Astron. Astrophys.}, {\bf 105}, pp. 359--361.

\item{}\kern-2pc
Aoki, S. and Kinoshita, H., 1983, ``Note on the relation between the
     equinox and Guinot's non-rotating origin," {\it Celest. Mech.},
     {\bf 29}, pp. 335--360.

\item{}\kern-2pc
Capitaine, N., 1990, ``The Celestial Pole Coordinates," {\it Celest.
     Mech. Dyn. Astr.}, {\bf 48}, pp. 127--143.
  
\item{}\kern-2pc
Capitaine, N., Guinot, B., and Souchay, J., 1986, ``A Non-rotating
     Origin on the Instantaneous Equator:  Definition, Properties and
     Use," {\it Celest. Mech.}, {\bf 39}, pp. 283--307.

\item{}\kern-2pc
Capitaine, N. and Chollet, F., 1991, ``The use of the nonrotating
     origin in the computation of apparent places of stars for
     estimating Earth Rotation Parameters," in {\it Reference Systems},
     J. A. Hughes, C. A. Smith, and G. H. Kaplan (eds), pp. 224--227.

\item{}\kern-2pc
Capitaine, N. and Gontier A. -M., 1991, ``Procedure for VLBI estimates
     of Earth Rotation Parameter referred to the nonrotating
     origin," in {\it Reference Systems}, J. A. Hughes, C. A. Smith, and
     G. H. Kaplan (eds), pp. 77--84.

\item{}\kern-2pc
Capitaine, N. and Gontier A. -M., 1993, ``Accurate procedure for
     deriving UT1 at a submilliarcsecond accuracy from Greenwich
     Sidereal Time or from stellar angle," {\it Astron. Astrophys.},
     {\bf 275}, pp. 645--650.

\item{}\kern-2pc
Defraigne, P., Dehant, V., and P\^aquet, P., 1995, ``Link between 
     the retrograde-prograde nutations and nutations in obliquity and 
     longitude," {\it Celest. Mech. Dyn. Astr.}, {\bf 62}, pp. 363--376.

\item{}\kern-2pc
Fukushima, T., 1991, ``Geodesic Nutation," {\it Astron. Astrophys.},
     {\bf 244}, pp. L11--L12. 

\item{}\kern-2pc
Gilbert, F. and Dziewonski, A. M., 1975, ``An Application of Normal
     Mode Theory to the Retrieval Structure Parameters and Source
     Mechanisms from Seismic Spectra," {\it Phil. Trans. Roy. Soc.
     Lond.}, {\bf A278}, pp. 187--269.

\item{}\kern-2pc
Gontier, A. -M., 1990, ``Tests pour l'application de l'origine
     nontournante en radioastronomie," in {\it Journ\'ees 1990
     Syst\`emes de R\'ef\'erence spatio-temporels}, pp. 281--286.

\item{}\kern-2pc
Guinot, B., 1979, ``Basic Problems in the Kinematics of the Rotation
     of the Earth," in {\it Time and the Earth's Rotation}, D. D.
     McCarthy and J. D. Pilkington (eds), D. Reidel Publishing
     Company, pp. 7--18.

\item{}\kern-2pc
Herring, T., 1996, Private Communication.

\item{}\kern-2pc
Kinoshita, H., 1977, ``Theory of the Rotation of the Rigid Earth,"
     {\it Celest. Mech.}, {\bf 15}, pp. 277--326.

\item{}\kern-2pc
Kinoshita, H. and Souchay, J., 1990, ``The Theory of the Nutation for
     the Rigid Earth Model at the Second Order," {\it Celest. Mech.
     and Dyn. Astr.}, {\bf 48}, pp. 187--265.

\item{}\kern-2pc
Lieske, J. H., Lederle, T., Fricke, W., and Morando, B., 1977,
     ``Expression for the Precession Quantities Based upon the IAU
     (1976) System of Astronomical Constants," {\it Astron. Astrophys.},
     {\bf 58}, pp. 1--16.

\item{}\kern-2pc
Lieske, J., 1991, Private Communication.

\item{}\kern-2pc
Mueller, I. I., 1969, {\it Spherical and Practical Astronomy as applied
     to Geodesy}, F. Ungar Publishing Co., Inc.

\item{}\kern-2pc
Seidelmann, P. K., 1982, ``1980 IAU Nutation:  The Final Report of
     the IAU Working Group on Nutation," {\it Celest. Mech.}, {\bf 27},
     pp. 79--106.

\item{}\kern-2pc
Simon, J. L., Bretagnon, P., Chapront, J., Chapront-Touz\'e, M.,
     Francou, G., Laskar, J., 1994, ``Numerical Expressions for 
     Precession Formulae and Mean Elements for the Moon and Planets,"
     {\it Astron. Astrophys.}, {\bf 282}, pp. 663--683.

\item{}\kern-2pc
Wahr, J. M., 1981, ``The Forced Nutations of an Elliptical, Rotating,
     Elastic, and Oceanless Earth," {\it Geophys. J. Roy. Astron.
     Soc.}, {\bf 64}, pp. 705--727.

\vfil\eject
